This suggests that month 1 is usually 95% of the trend, month 2 is 121% and month 3 is 84%. The multiplicative model is a better method to use when the trend is increasing or decreasing over time, as the seasonal variation is also likely to be increasing or decreasing.

Note that with the additive model the three seasonal variations must add up to zero (32-25-7 = 0). Where this is not the case, an adjustment must be made. With the multiplicative model the three seasonal variations add to three (0.95 + 1.21 + 0.84 = 3). (If it was four-month average, the four seasonal variations would add to four etc). Again, if this is not the case, an adjustment must be made.

In this simplified example the trend shows an increase of exactly $2,000 each month, and the pattern of seasonal variations is exactly the same in each three-month period. In reality a time series is unlikely to give such a perfect result.

#### Step 5 – Using time series to forecast the future

Now that the trend and the seasonal variations have been calculated, these can be used to predict the likely level of sales revenue for the future.

**Question:**

Using the above example, what is the predicted level of sales revenue for June 20X3 and July 20X3?

**Solution:**

Start with the trend then apply the seasonal variations. We calculated an increasing trend of $2,000 per month. The last figure we calculated was for November 20X2 showing $170,000. If we assume the trend continues as it has done previously, then by June 20X3, the sales revenue figure will have increased by $14,000 ($2,000 per month for seven months). Adding this to the figure we have for November, we can predict the underlying trend value for June 20X3 to be $184,000. ($14,000 + $170,000).

We know that sales exhibit a seasonal variation. Taking account of the seasonal variation will give us a better estimate for June 20X3. From the table in step 4, we can see that June has a positive variation of $32,000.

Our estimate for the sales revenue for June 20X3 is therefore $184,000 + $32,000 = $216,000.

For July, the underlying trend value will be $170,000 + $16,000 = $186,000. The seasonal variation for July 20X3 is a negative variation of $25,000, therefore our estimate for the sales revenue for July 20X3 is $186,000 - $25,000 = $161,000.

**Calculating moving averages for an even number of periods**

In the above example, we used a three-month moving average. Looking back at step 2, we can see that the average is shown against the mid-point of the three observations. The mid-point of the period for January, February and March is shown against the February observation.

When we are calculating a moving average with an even number of periods, for example **a four-quarter moving average, **we do the same basic calculation, but the mid-point will lie between observations. From step 4 above, we can see that we need the moving average to be shown against an observation so that the seasonal variation can be calculated. We therefore calculate the four-quarter moving average as before, but we then calculate a second moving average.

In the example below, the four-quarter moving averages have been calculated in the same way as before. The first four observations are added together and then divided by four. The four-quarter moving average for the first four quarters is 322.50. Moving to the next four observations, gives an average of 327.50. We can then work out the mid-point of these two averages by adding them together and dividing by two. This gives a mid-point of (322.50 + 327.50) ÷ 2 = 325. This mid-point is our trend and the figure is shown against the quarter 3, 20X8 observation. All other calculations are done in the same way as our original example.